The range of stable random walks. (English) Zbl 0729.60066

Le Gall, Jean-François; Rosen, Jay

Ann. Probab. 19, No. 2, 650-705 (1991).

Let \(X_ n=x_ 0+\sum^{n}_{i}Y_ i\), where \(x_ 0\) and \((Y_ i\), \(i\geq 1)\) are in the d-dimensional integer lattice \({\mathbb{Z}}^ d\), describe a d-dimensional random walk. The range \(R_ n\) of the random walk is the number of distinct sites visited by the random walk up to time n. The basic assumption is that \[ b(n)^{-1}X_ n\begin{matrix} (d)\\ n\to \infty \end{matrix} U_ 1, \] where the convergence is in distribution, b(n) a regularly varying sequence of index 1/\(\beta\), \(\beta\in (0,2]\) and \(U=(U_ t\), \(t>0)\) a nondegenerate stable process of index \(\beta\) in \({\mathbb{R}}^ d.\)
This important paper treats laws of large numbers and central limit theorems for \(R_ n\). The authors use a wealth of analytic and probabilistic tools to prove the following results:
(i) If \(\beta\leq 2d/3\) (transient case), \(R_ n/n\to q\) a.s. where \(q=P(X_ 1\neq x_ 0\), \(X_ 2\neq x_ 0,...)\) and \([R_ n-E(R_ n)]/\sqrt{ng(n)}\) converges to \(\sigma\) N with N a standard normal, \(\sigma\) an explicit constant and \(g(n)=\sum^{n}_{1}k^ 2b(k)^{- 2d}.\)
(ii) If \(2d/3<\beta \leq d\) (transient or recurrent), \(R_ nh(n)/n\to 1\) a.s. where \(h(n)=\sum^{n}_{0}P(X_ k=X_ 0)\); further \(n^{- 2}h(n)^ 2b(n)^ d[R_ n-E R_ n]\to^{(d)}-\gamma_ U,\) where \(\gamma_ U\) is a renormalized self-intersection local time of the process U.
(iii) If \(\beta >d\) (hence \(d=1)\), \(b(n)^{-1}R_ n\to^{(d)}m(\{U_ s\); \(0\leq s\leq 1\})\) where m is Lebesgue measure on \({\mathbb{R}}.\)
The current paper not only connects a number of previous results by e.g. Jain, Pruitt, and Taylor but adds a vast set of new limit theorems. Some results require regularity assumptions on the characteristic function of X.

Reviewer: J.L.Teugels (Heverlee)

MSC:

60G50	Sums of independent random variables; random walks
60F05	Central limit and other weak theorems
60E07	Infinitely divisible distributions; stable distributions
60E10	Characteristic functions; other transforms
60F17	Functional limit theorems; invariance principles
60J55	Local time and additive functionals

Keywords:

range of random walk; number of distinct sites visited by the random walk; regularly varying sequence; laws of large numbers; central limit theorems; self-intersection local time